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3818A generalization of Pascal theorem and Pappus theorem

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  • yeuemtrondoitb85
    Jun 18, 2017

      Dear Dr. Paul Yiu, and Geometers,


      Let 1, 2, 3, 4, 5, 6 be six arbitrary points in a hyperbola. Let 1' be arbitrary point in the hyperbola. The circle (121') meets the hyperbola at point 2'. The circle (232) meets the hyperbola again at 3', define points 4', 5', 6' similarly. Let circle (121') meets the circle (454') at A, B, Let circle (232') meets the circle (565') at C, D. Let circle (343') meets the circle (616')  at E, F. Then six points A, B, C, D, E, F lie on a circle.


      Special case:


      1. If 1' at \infty the theorem is Pascal theorem


      https://en.wikipedia.org/wiki/Pascal%27s_theorem


      2. If the hyperbola is two lines, and 1' at \infty the theorem is Pappus theorem


      https://en.wikipedia.org/wiki/Pappus%27s_hexagon_theorem


      Best regards

      Sincerely
      Dao Thanh Oai